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G = Q8×Dic5order 160 = 25·5

Direct product of Q8 and Dic5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8×Dic5, C54(C4×Q8), (C5×Q8)⋊6C4, C2.3(Q8×D5), (C2×Q8).5D5, C20.35(C2×C4), (C2×C4).56D10, (Q8×C10).5C2, C10.16(C2×Q8), C4.4(C2×Dic5), C4⋊Dic5.12C2, (C4×Dic5).4C2, C10.35(C4○D4), (C2×C20).39C22, C10.39(C22×C4), (C2×C10).57C23, C2.3(Q82D5), C2.7(C22×Dic5), C22.26(C22×D5), (C2×Dic5).43C22, SmallGroup(160,166)

Series: Derived Chief Lower central Upper central

C1C10 — Q8×Dic5
C1C5C10C2×C10C2×Dic5C4×Dic5 — Q8×Dic5
C5C10 — Q8×Dic5
C1C22C2×Q8

Generators and relations for Q8×Dic5
 G = < a,b,c,d | a4=c10=1, b2=a2, d2=c5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 152 in 70 conjugacy classes, 51 normal (14 characteristic)
C1, C2 [×3], C4 [×6], C4 [×5], C22, C5, C2×C4 [×3], C2×C4 [×4], Q8 [×4], C10 [×3], C42 [×3], C4⋊C4 [×3], C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×6], C2×C10, C4×Q8, C2×Dic5, C2×Dic5 [×3], C2×C20 [×3], C5×Q8 [×4], C4×Dic5 [×3], C4⋊Dic5 [×3], Q8×C10, Q8×Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, D5, C22×C4, C2×Q8, C4○D4, Dic5 [×4], D10 [×3], C4×Q8, C2×Dic5 [×6], C22×D5, Q8×D5, Q82D5, C22×Dic5, Q8×Dic5

Smallest permutation representation of Q8×Dic5
Regular action on 160 points
Generators in S160
(1 48 28 34)(2 49 29 35)(3 50 30 36)(4 41 21 37)(5 42 22 38)(6 43 23 39)(7 44 24 40)(8 45 25 31)(9 46 26 32)(10 47 27 33)(11 131 151 142)(12 132 152 143)(13 133 153 144)(14 134 154 145)(15 135 155 146)(16 136 156 147)(17 137 157 148)(18 138 158 149)(19 139 159 150)(20 140 160 141)(51 71 65 82)(52 72 66 83)(53 73 67 84)(54 74 68 85)(55 75 69 86)(56 76 70 87)(57 77 61 88)(58 78 62 89)(59 79 63 90)(60 80 64 81)(91 122 102 111)(92 123 103 112)(93 124 104 113)(94 125 105 114)(95 126 106 115)(96 127 107 116)(97 128 108 117)(98 129 109 118)(99 130 110 119)(100 121 101 120)
(1 68 28 54)(2 69 29 55)(3 70 30 56)(4 61 21 57)(5 62 22 58)(6 63 23 59)(7 64 24 60)(8 65 25 51)(9 66 26 52)(10 67 27 53)(11 122 151 111)(12 123 152 112)(13 124 153 113)(14 125 154 114)(15 126 155 115)(16 127 156 116)(17 128 157 117)(18 129 158 118)(19 130 159 119)(20 121 160 120)(31 82 45 71)(32 83 46 72)(33 84 47 73)(34 85 48 74)(35 86 49 75)(36 87 50 76)(37 88 41 77)(38 89 42 78)(39 90 43 79)(40 81 44 80)(91 142 102 131)(92 143 103 132)(93 144 104 133)(94 145 105 134)(95 146 106 135)(96 147 107 136)(97 148 108 137)(98 149 109 138)(99 150 110 139)(100 141 101 140)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)
(1 91 6 96)(2 100 7 95)(3 99 8 94)(4 98 9 93)(5 97 10 92)(11 90 16 85)(12 89 17 84)(13 88 18 83)(14 87 19 82)(15 86 20 81)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 134 56 139)(52 133 57 138)(53 132 58 137)(54 131 59 136)(55 140 60 135)(61 149 66 144)(62 148 67 143)(63 147 68 142)(64 146 69 141)(65 145 70 150)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)

G:=sub<Sym(160)| (1,48,28,34)(2,49,29,35)(3,50,30,36)(4,41,21,37)(5,42,22,38)(6,43,23,39)(7,44,24,40)(8,45,25,31)(9,46,26,32)(10,47,27,33)(11,131,151,142)(12,132,152,143)(13,133,153,144)(14,134,154,145)(15,135,155,146)(16,136,156,147)(17,137,157,148)(18,138,158,149)(19,139,159,150)(20,140,160,141)(51,71,65,82)(52,72,66,83)(53,73,67,84)(54,74,68,85)(55,75,69,86)(56,76,70,87)(57,77,61,88)(58,78,62,89)(59,79,63,90)(60,80,64,81)(91,122,102,111)(92,123,103,112)(93,124,104,113)(94,125,105,114)(95,126,106,115)(96,127,107,116)(97,128,108,117)(98,129,109,118)(99,130,110,119)(100,121,101,120), (1,68,28,54)(2,69,29,55)(3,70,30,56)(4,61,21,57)(5,62,22,58)(6,63,23,59)(7,64,24,60)(8,65,25,51)(9,66,26,52)(10,67,27,53)(11,122,151,111)(12,123,152,112)(13,124,153,113)(14,125,154,114)(15,126,155,115)(16,127,156,116)(17,128,157,117)(18,129,158,118)(19,130,159,119)(20,121,160,120)(31,82,45,71)(32,83,46,72)(33,84,47,73)(34,85,48,74)(35,86,49,75)(36,87,50,76)(37,88,41,77)(38,89,42,78)(39,90,43,79)(40,81,44,80)(91,142,102,131)(92,143,103,132)(93,144,104,133)(94,145,105,134)(95,146,106,135)(96,147,107,136)(97,148,108,137)(98,149,109,138)(99,150,110,139)(100,141,101,140), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,91,6,96)(2,100,7,95)(3,99,8,94)(4,98,9,93)(5,97,10,92)(11,90,16,85)(12,89,17,84)(13,88,18,83)(14,87,19,82)(15,86,20,81)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(31,114,36,119)(32,113,37,118)(33,112,38,117)(34,111,39,116)(35,120,40,115)(41,129,46,124)(42,128,47,123)(43,127,48,122)(44,126,49,121)(45,125,50,130)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,149,66,144)(62,148,67,143)(63,147,68,142)(64,146,69,141)(65,145,70,150)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155)>;

G:=Group( (1,48,28,34)(2,49,29,35)(3,50,30,36)(4,41,21,37)(5,42,22,38)(6,43,23,39)(7,44,24,40)(8,45,25,31)(9,46,26,32)(10,47,27,33)(11,131,151,142)(12,132,152,143)(13,133,153,144)(14,134,154,145)(15,135,155,146)(16,136,156,147)(17,137,157,148)(18,138,158,149)(19,139,159,150)(20,140,160,141)(51,71,65,82)(52,72,66,83)(53,73,67,84)(54,74,68,85)(55,75,69,86)(56,76,70,87)(57,77,61,88)(58,78,62,89)(59,79,63,90)(60,80,64,81)(91,122,102,111)(92,123,103,112)(93,124,104,113)(94,125,105,114)(95,126,106,115)(96,127,107,116)(97,128,108,117)(98,129,109,118)(99,130,110,119)(100,121,101,120), (1,68,28,54)(2,69,29,55)(3,70,30,56)(4,61,21,57)(5,62,22,58)(6,63,23,59)(7,64,24,60)(8,65,25,51)(9,66,26,52)(10,67,27,53)(11,122,151,111)(12,123,152,112)(13,124,153,113)(14,125,154,114)(15,126,155,115)(16,127,156,116)(17,128,157,117)(18,129,158,118)(19,130,159,119)(20,121,160,120)(31,82,45,71)(32,83,46,72)(33,84,47,73)(34,85,48,74)(35,86,49,75)(36,87,50,76)(37,88,41,77)(38,89,42,78)(39,90,43,79)(40,81,44,80)(91,142,102,131)(92,143,103,132)(93,144,104,133)(94,145,105,134)(95,146,106,135)(96,147,107,136)(97,148,108,137)(98,149,109,138)(99,150,110,139)(100,141,101,140), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,91,6,96)(2,100,7,95)(3,99,8,94)(4,98,9,93)(5,97,10,92)(11,90,16,85)(12,89,17,84)(13,88,18,83)(14,87,19,82)(15,86,20,81)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(31,114,36,119)(32,113,37,118)(33,112,38,117)(34,111,39,116)(35,120,40,115)(41,129,46,124)(42,128,47,123)(43,127,48,122)(44,126,49,121)(45,125,50,130)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,149,66,144)(62,148,67,143)(63,147,68,142)(64,146,69,141)(65,145,70,150)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155) );

G=PermutationGroup([(1,48,28,34),(2,49,29,35),(3,50,30,36),(4,41,21,37),(5,42,22,38),(6,43,23,39),(7,44,24,40),(8,45,25,31),(9,46,26,32),(10,47,27,33),(11,131,151,142),(12,132,152,143),(13,133,153,144),(14,134,154,145),(15,135,155,146),(16,136,156,147),(17,137,157,148),(18,138,158,149),(19,139,159,150),(20,140,160,141),(51,71,65,82),(52,72,66,83),(53,73,67,84),(54,74,68,85),(55,75,69,86),(56,76,70,87),(57,77,61,88),(58,78,62,89),(59,79,63,90),(60,80,64,81),(91,122,102,111),(92,123,103,112),(93,124,104,113),(94,125,105,114),(95,126,106,115),(96,127,107,116),(97,128,108,117),(98,129,109,118),(99,130,110,119),(100,121,101,120)], [(1,68,28,54),(2,69,29,55),(3,70,30,56),(4,61,21,57),(5,62,22,58),(6,63,23,59),(7,64,24,60),(8,65,25,51),(9,66,26,52),(10,67,27,53),(11,122,151,111),(12,123,152,112),(13,124,153,113),(14,125,154,114),(15,126,155,115),(16,127,156,116),(17,128,157,117),(18,129,158,118),(19,130,159,119),(20,121,160,120),(31,82,45,71),(32,83,46,72),(33,84,47,73),(34,85,48,74),(35,86,49,75),(36,87,50,76),(37,88,41,77),(38,89,42,78),(39,90,43,79),(40,81,44,80),(91,142,102,131),(92,143,103,132),(93,144,104,133),(94,145,105,134),(95,146,106,135),(96,147,107,136),(97,148,108,137),(98,149,109,138),(99,150,110,139),(100,141,101,140)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160)], [(1,91,6,96),(2,100,7,95),(3,99,8,94),(4,98,9,93),(5,97,10,92),(11,90,16,85),(12,89,17,84),(13,88,18,83),(14,87,19,82),(15,86,20,81),(21,109,26,104),(22,108,27,103),(23,107,28,102),(24,106,29,101),(25,105,30,110),(31,114,36,119),(32,113,37,118),(33,112,38,117),(34,111,39,116),(35,120,40,115),(41,129,46,124),(42,128,47,123),(43,127,48,122),(44,126,49,121),(45,125,50,130),(51,134,56,139),(52,133,57,138),(53,132,58,137),(54,131,59,136),(55,140,60,135),(61,149,66,144),(62,148,67,143),(63,147,68,142),(64,146,69,141),(65,145,70,150),(71,154,76,159),(72,153,77,158),(73,152,78,157),(74,151,79,156),(75,160,80,155)])

Q8×Dic5 is a maximal subgroup of
Dic5.12Q16  Dic57SD16  C5⋊Q165C4  Dic54Q16  Q8⋊Dic10  Dic5.9Q16  Q8.Dic10  Q8.2Dic10  Q8⋊D56C4  Dic55SD16  SD16⋊Dic5  (C5×Q8).D4  Dic53Q16  Q16⋊Dic5  (C2×Q16)⋊D5  C20.6M4(2)  Q85Dic10  Q86Dic10  C4×Q8×D5  C42.125D10  C4×Q82D5  C42.126D10  (Q8×Dic5)⋊C2  C22⋊Q825D5  C10.152- 1+4  C10.1182+ 1+4  C10.212- 1+4  C10.232- 1+4  C10.772- 1+4  C10.242- 1+4  C42.139D10  C42.234D10  C42.143D10  C42.144D10  Dic108Q8  C42.241D10  C42.176D10  C42.177D10  C10.422- 1+4  C10.452- 1+4  C10.1062- 1+4  C10.1072- 1+4  C10.1482+ 1+4  Dic157Q8
Q8×Dic5 is a maximal quotient of
C4⋊C45Dic5  C206(C4⋊C4)  C42.210D10  (Q8×C10)⋊17C4  Dic157Q8

40 conjugacy classes

class 1 2A2B2C4A···4F4G4H4I4J4K···4P5A5B10A···10F20A···20L
order12224···444444···45510···1020···20
size11112···2555510···10222···24···4

40 irreducible representations

dim111112222244
type++++-++--+
imageC1C2C2C2C4Q8D5C4○D4D10Dic5Q8×D5Q82D5
kernelQ8×Dic5C4×Dic5C4⋊Dic5Q8×C10C5×Q8Dic5C2×Q8C10C2×C4Q8C2C2
# reps133182226822

Matrix representation of Q8×Dic5 in GL5(𝔽41)

400000
01000
00100
00001
000400
,
400000
01000
00100
0002038
0003821
,
400000
00100
040600
00010
00001
,
320000
0162500
0392500
00010
00001

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,1,0],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,20,38,0,0,0,38,21],[40,0,0,0,0,0,0,40,0,0,0,1,6,0,0,0,0,0,1,0,0,0,0,0,1],[32,0,0,0,0,0,16,39,0,0,0,25,25,0,0,0,0,0,1,0,0,0,0,0,1] >;

Q8×Dic5 in GAP, Magma, Sage, TeX

Q_8\times {\rm Dic}_5
% in TeX

G:=Group("Q8xDic5");
// GroupNames label

G:=SmallGroup(160,166);
// by ID

G=gap.SmallGroup(160,166);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,188,86,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^10=1,b^2=a^2,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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